Fourier spectra and benchmarking of phase-estimation variational quantum classifiers

Introduction: A variational quantum classifier realises a function whose Fourier spectrum is fixed by the circuit and whose coefficients must be found by an optimiser. We study a readout borrowed from quantum phase estimation (QPE): a data-dependent operator acts on a target register through controlled powers, and the class label is read from an estimation register of m qubits after an inverse quantum Fourier transform (QFT). Materials and methods: We derive the exact readout distribution—a Fejér-weighted expansion in the moments of the operator—and characterise its Fourier spectrum for two single-qubit blocks: H ⋅ P h ⋅ R Y (family A) and a universal U 2 block (family B). All analytical statements are verified against a state-vector simulator, including attainment of the predicted spectrum. Three ablation controls alter one component each at an unchanged parameter count: the encoding QFT is removed, the exponential ladder of controlled powers is replaced by single powers, and the inverse QFT is replaced by independent Hadamard tests. The benchmark comprises 7680 training runs of 24 circuits on three tasks, with one generated dataset or one cross-validation fold as the unit of inference (twenty per sample size), exact signed-rank tests, and the corrected resampled t -test for the overlapping folds. Results: The support of every readout probability lies in parity-locked lattice cubes of maximal harmonic order 2 m + 1 . The maximal order is attained at every register size, in every feature and every outcome, except for family A with one target qubit, where the two highest orders cancel identically; the full support is attained at every register size used here ( t ≤ 4 , m ≤ 3 ), and the band is tapered towards its edge. Removing the encoding QFT costs family B 0.144 and 0.082 in test accuracy on the two-feature tasks, and replacing the power ladder by single powers costs 0.086 – 0.225 wherever m ≥ 2 ; replacing the inverse QFT by Hadamard tests has no detectable effect. The architecture leads the independently parameterised reference circuits only on the three-class task ( + 0.119 , p < 0.0001 ), where a parameter-tied re-uploading circuit with 12 rather than 20 parameters is not resolved from it ( + 0.023 , 95% confidence interval (CI)